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John Baez (Oct 14 2025 at 14:41):Have any you looked at this paper by Hemelaer defining the dimension of a Grothendieck topos?
Chris Grossack (she/they) (Oct 14 2025 at 14:46):No but that sounds really interesting! I asked a question about this on mse a while ago. I'm excited to see what Jens came up with!
Morgan Rogers (he/him) (Oct 14 2025 at 15:12):I have looked at the abstract at least, which seems very interesting.
Jens Hemelaer (Oct 14 2025 at 19:53):Hi! It's been a while since I was last active here (in part because of that paper :smile:), but I'll keep an eye on Zulip in case there are any questions/comments (thanks Morgan for letting me know about the thread!).
John Baez (Oct 14 2025 at 19:57):I wish someone would explain the ideas in a simple way, but that's because I'm lazy and also a bit scared to read the paper. If I were writing about it in This Week's Finds I'd try to understand it. I'm particularly curious about the factor of 2 in the abstract, which reminds me of the factor of 2 that comes up in etale cohomology.
Jens Hemelaer (Oct 14 2025 at 20:07):The basic idea is that (for spaces of dimension ), locally, removing a point changes the cohomology in degree , but does not change the cohomology in degrees .
Jens Hemelaer (Oct 14 2025 at 20:09):By extending it to all toposes you also get a dimension theory for the petit étale toposes of schemes, and there the relevant cohomology groups are étale cohomology groups. The calculation there uses existing results for étale cohomology.
Jens Hemelaer (Oct 14 2025 at 20:28):I had to look it up again, but I rely heavily on existing cohomological purity results here in étale cohomology. I don't have a very good intuition about the factor 2, other than that it makes sense if you work over the complex numbers... For example corresponding to the complex plane.
John Baez (Oct 14 2025 at 20:29):Great, thanks! I have a halfway decent intuition for the factor of 2 in etale cohomology, which is incredibly important, but I wondered how it got into your work on topos theory, so I'm glad to hear it does so via results on etale cohomology.